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The theorem which Abel thus briefly describes is to-day
known as Abel's Theorem. His proof of it has been described as
nothing more than 6a marvellous exercise in the integral
calculus'. As in his algebra, so in his analysis, Abel attained his
proof with a superb parsimony. The proof, it may be said with-
out exaggeration, is well within the purview of any seventeen-
year-old who has been through a good first course in the calcu-
lus. There is nothing high-falutin' about the classic simplicity
of Abel's own proof. The like cannot be said for some of the
nineteenth-century expansions and geometrical reworkings of
the original proof. Abel's proof is like a statue by Phidias; some
of the others resemble a Gothic cathedral smothered in Irish
lace, Italian confetti, and French pastry.
There is ground for a possible misunderstanding in Abel's
opening paragraph. Abel no doubt was merely being kindly
courteous to an old man who had patronized him - in the bad
sense - on first acquaintance, but who, nevertheless, had spent
most of his long working life on an important problem without
seeing what it was all about. It is not true that Legendre had
discussed the elliptic functions, as Abel's words might imply;
what Legendre spent most of his life over was elliptic integrals
which are as different from elliptic functions as a horse is from
the cart it pulls, and therein precisely is the crux and the germ
of one of Abel's greatest contributions to mathematics. The
matter is quite simple to anyone who has had a school course in
trigonometry; to obviate tedious explanations of elementary
matters this much will be assumed in what follows presently.
For those who have forgotten afl about trigonometry, how-
ever, the essence, the methodology, of Abel's epochal advance
can be analogized thus. "VYe alluded to the cart and the horse.
The frowsy proverb about putting the cart before the horse
describes what Legendie did; Abel saw that if the cart was to
move forward the horse should precede it. To take another
instance: Francis Galton, in his statistical studies of the relation
between poverty and chronic drunkenness, was led, by his
impartial mind, to a reconsideration of all the self-righteous
platitudes by which indignant moralists and economic crusaders
with an axe to grind evaluate such social phenomena. Instead